Logarithmic norms and regular perturbations of differential equations

Jacek Banasiak


In this note we explore the concept of the logarithmic norm of a matrix and illustrate its applicability by using it to find conditions under which the convergence of solutions of regularly perturbed systems of ordinary differential equations is uniform globally in time.


Logarithmic norm; stability; regular perturbations of differential equations

Full Text:



Carr, J., Applications of Centre Manifold Theory, Applied Mathematical Sciences 35, Springer-Verlag, New York–Berlin, 1981.

Coppel, W. A., Stability and Asymptotic Behavior of Differential Equations, D. C. Heath and Co., Boston, Mass., 1965.

Dahlquist, G., Stability and error bounds in the numerical integration of ordinary differential equations, Kungl. Tekn. H¨ogsk. Handl. Stockholm. No. 130 (1959), 87pp.

Engel, K.-J., Nagel, R., One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics 194, Springer-Verlag, New York, 2000.

Hairer, E., Nørsett, S. P., Wanner, G., Solving Ordinary Differential Equations. I. Nonstiff Problems, Springer Series in Computational Mathematics 8, Springer-Verlag, Berlin, second edition, 1993.

Lozinskii, S. M., Error estimate for numerical integration of ordinary differential equations. I, Izv. Vysˇs. Uˇcebn. Zaved. Matematika 5 (6) (1958), 52–90 (Russian); 5 (12) (1959), 222 (Erratum).

Marciniak-Czochra, A., Mikelic, A., Stiehl, T., Renormalization group second-order approximation for singularly perturbed nonlinear ordinary differential equations, Math. Methods Appl. Sci. 41 (14) (2018), 5691–5710.

Ortega, J. M., Numerical Analysis, Classics in Applied Mathematics 3, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, second edition, 1990.

Pao, C. V., A further remark on the logarithmic derivatives of a square matrix, Linear Algebra and Appl. 7 (1973), 275–278.

Pao, C. V., Logarithmic derivates of a square matrix, Linear Algebra and Appl. 6 (1973), 159–164.

Sallet, G., Mathematical Epidemiology, Lecture Notes, 2018. http://www.iecl.univ-lorraine.fr/Gauthier.Sallet/Lecture-Notes-Pretoria-2018.pdf

Smith, H. L., Waltman, P., The Theory of the Chemostat, Cambridge Studies in Mathematical Biology 13, Cambridge University Press, Cambridge, 1995.

Soderlind, G., The logarithmic norm. History and modern theory, BIT 46 (3) (2006), 631–652.

Strom, T., On logarithmic norms, SIAM J. Numer. Anal. 12 (5) (1975), 741–753.

Uherka, D. J., Sergott, A. M., On the continuous dependence of the roots of a polynomial on its coefficients, Amer. Math. Monthly 84 (5) (1977), 368–370.

Vladimirov, V. S., Equations of Mathematical Physics, Pure and Applied Mathematics 3, Marcel Dekker, Inc., New York, 1971.

Walter, W., Ordinary Differential Equations, Graduate Texts in Mathematics 182, Springer-Verlag, New York, 1998.

DOI: http://dx.doi.org/10.17951/a.2019.73.2.5-19
Data publikacji: 2020-01-16 07:29:30
Data złożenia artykułu: 2019-12-29 16:23:32


  • There are currently no refbacks.

Copyright (c) 2019 Jacek Banasiak